Research theorem: sharp scratch-dependent call bound for dirty echoes

[AlexS-scale]

Proves a sharp call bound for ordered dirty-echo trees: N >= 2^(s+1)n - 4^s, with an explicit attaining family for every s >= 2 and 3*2^(s-2) <= n <= 2^s. In particular, 50 factors with six arbitrary dirty arrays have exact minimum 2,304 complete leaf calls, versus 2,752 for the pinned balanced binary word. All inverse and restoration work is included.

Status: research note / independent verification; no new kappa. The theorem is scoped to static ordered echo trees with reserved parent scratch. It is not a minimum over arbitrary reversible programs or a multiplication improvement.

Concurrent #758 already supplies all-arity weighted optimization; #730 supplies mixed binary/ternary optimization and #744 a strict-ternary audit. This contribution credits them and adds the analytic scratch-dependent bound and equality family. Its independent uniform-cost implementation also corroborates #758 at nine pinned instances. The echo itself is #673's, the stronger balanced comparator is #689's, and the rounded leaves are #628's.

Proof · Exact certificate · Source pins · Validation

python3 -I -O -B research/dirty-echo-space-budget/verify.py

Expected: PASS_DIRTY_ECHO_SPACE_BUDGET: n=50, arrays=6, leaves=2752->2304.

Validation passes on macOS arm64 / Python 3.11.15 and 3.14.7:

• All 576 declared states, 255 literal witnesses, 5,440 independently enumerated small trees, and 36 explicit equality-family constructions.
• 2,904 signed noncommutative identities/inverse checks, 1,530 nonlinear round trips, and 186 rounded compositions with dense exact references and counted phase reads.
• Instruction and interface negative controls, relocated replay, seven artifact-corruption/overwrite controls, 13 exact Git source bindings, and selected-record integrity.

One isolated make verify completed seven historical groups, then stopped at gen4-bank-verify because SymPy is absent. No full-suite, supplier cold-replay, remote CI, Lean or independent human-review pass is claimed.

Exact base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. OS assignment seed 50, recursive interfaces. Initial claim was local because no pinned coordination issue existed. Final full refresh covered 586 open PRs through #767; the immediate pre-submission check covered arrivals through #782. PR673 advanced to b3917936aafbd0d680a954a86e25c4ce7e28a9f9; both consumed files remain byte-identical to the pinned head. The selected frontier and upstream/ are unchanged.

Prepared by Alexander Schwartzman (@AlexS-scale) with substantial OpenAI Codex assistance, under Apache-2.0. Source notices and concurrent credit are retained. Uniform/compiler, physical hosting, precision, tape and analytic obligations remain inherited; finite replay is not a formal proof of the multiplication theorem.

[Mister-Meeseeks]

100-PR batch review at 548e42022b37a300f8bf745e848c38baccbce056.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Static dirty-echo scratch/call tradeoff with uniform lower bound and attaining interval; reserved connector grammar only.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.